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Integrating supplier selection decisions into an inventory location problem for designing the supply chain network

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Abstract

This paper proposes a novel Inventory-Location problem that integrates supplier selection decisions to design a three-echelon supply chain network, under a continuous (s,Q) inventory control policy at the warehouses. In this problem, a set of warehouses must be selected within a set of potential locations to serve several customers or demand zones, additionally involving the selection of the suppliers for fulfilling incoming orders from the located warehouses. The optimal solution must be determined while minimizing total system costs including supplier selection, transportation (i.e., suppliers-warehouses and warehouses-customers), inventory (i.e., cycle and safety stock), and warehouse location costs. A key element of the problem is the consideration of variable lead-times for the warehouses, which are dependent on the selection of the supplier that serve them, thus increasing model complexity. Accordingly, an efficient algorithm based on the Generalized Benders Decomposition is developed and implemented to solve the proposed Mixed Integer, Nonlinear, Nonconvex, Programming Model. The proposed solution approach relies on a convenient model formulation and decomposition that yields a Mixed Integer Linear master problem and a continuous, convex subproblem. A wide set of medium-sized synthetic instances are optimally solved in affordable times, denoting the efficiency and effectiveness of the proposed model along with the proposed solution approach. Significant scientific and managerial insights are provided and discussed.

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Acknowledgements

This research work was performed by ANID FONDECYT grant number 1123046. In addition, this research was also supported by “Proyecto Iniciación a la Investigación 2022” of Universidad Católica del Norte grant number VRIDT No. 055/2022. Furthermore, this research work was supported partially by “Postdoctorado DI 2019” of Pontificia Universidad Católica de Valparaíso, Grant Agreement Number 37.0/2019.

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Authors and Affiliations

  1. Department of Industrial Engineering, Universidad Católica del Norte, 1270709, Antofagasta, Chile

    Francisco J. Tapia-Ubeda & Pablo A. Miranda-Gonzalez

  2. School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, 2362807, Valparaíso, Chile

    Gabriel Gutiérrez-Jarpa

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  1. Francisco J. Tapia-Ubeda
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Correspondence to Francisco J. Tapia-Ubeda.

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Appendices

Appendix A

This appendix provides a simple analysis to demonstrate the convenience of selecting a single supplier instead of two suppliers for serving a warehouse with known demand mean and variance. For this demonstration, we assume that, in cases of employing two suppliers with different lead times, the perceived lead time at the warehouse would be the average value, considering the proportion of demand served by each supplier as weighing parameter. Any other assumption may potently increase the perceived lead time at the warehouse, for example integrating the lead time variance, or considering the maximum lead time of the two values, yielding more advantages of selecting a single supplier.

Following the notation in Sect. 3, omitting the index i of the warehouse, and considering that θ is the proportion of demand served by supplier 2 respect to the demand served by supplier 1, the total costs perceived by the warehouse can be written as follows (all remaining costs does not depend on supplier selection):

$$\begin{aligned} G\left( \theta \right) & = HC \cdot Z_{1 - \alpha } \sqrt {\left( {\left( {1 - \theta } \right) \cdot LT_{1} + \theta \cdot LT_{2} } \right)} \cdot \sqrt V \\ & + CIn_{1} \cdot \left( {1 - \theta } \right) \cdot D + CIn_{2} \cdot \theta \cdot D \end{aligned}$$
(38)

This analysis aims at demonstrating that selecting one supplier (θ = 0 or θ = 1) always yield a lower cost compared to operating the two of them (0 < θ < 1). Without loss of generality, we assume that total cost of employing suppler 1 \((\theta = 0)\) is lower than selecting supplier 2 \((\theta = 1)\), i.e., the following expression holds.

$$HC \cdot Z_{1 - \alpha } \sqrt {LT_{1} } \cdot \sqrt V + CIn_{1} \cdot D < HC \cdot Z_{1 - \alpha } \sqrt {LT_{2} } \cdot \sqrt V + CIn_{2} \cdot D$$
(39)

The second derivative of G(θ) the expression is:

$$\frac{{\partial G^{2} \left( \theta \right)}}{{\partial \theta^{2} }} = - \frac{1}{4}HC \cdot Z_{1 - \alpha } \cdot \frac{{\left( {LT_{2} - LT_{1} } \right)^{2} }}{{\left( {\left( {1 - \theta } \right) \cdot LT_{1} + \theta \cdot LT_{2} } \right)^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }} \cdot \sqrt V$$
(40)

Which is allays negative when LT1 ≠ LT2 and the costs function is strictly concave. Accordingly, any value of 0 < θ < 1 will always provide a more expensive cost than θ = 0 and selecting suppler 1 is always better than employing the two suppliers.

Notice that:

  • When the lead times are the same, the selection would be driven by the transportation costs, and again one supplier will be better than two suppliers (except when the transportation costs are the same, where choosing one suppler is also optimum).

  • If fixed cost of selecting suppliers are added in to the analysis, the costs function when 0 < θ < 1 is a vertical shift up, which does not affect the obtained result.

Appendix B

In this appendix, a numerical example is presented considering 2 suppliers, 5 potential warehouses, and 5 customers, for which the proposed formulation is solved by using the proposed GBD-based algorithm. In this case, after integrating the five initial cuts (P = 1, 2, 3, 4, 5), the algorithm employed 8 iterations for obtaining the final optimal solution. The MP solutions for all iterations are presented in Fig. 

Fig. 30
figure 30

Solutions obtained with the MP at each iteration

Full size image

30 (except for the iteration 4). In addition, Table 9 and Fig. 31 show information about solutions obtained at each iteration.

Table 9 Objective function values for each iteration
Full size table
Fig. 31
figure 31

Objective function evolution

Full size image

The next is the set of cuts integrated into the MP, including the first five cuts obtained with the artificial or modified MP (i.e., P = 1, 2, 3, 4, 5), and the seven cuts obtained after solving the respective SP at each iteration.

figure a

Appendix C

 

Base Instance 1

Base Instance 2

#Customers (nC)

#Customers (nC)

5

10

15

20

5

10

15

20

#Suppliers (nS) = 1

#Warehouses (nW) = 5

OF

1,653,895.92

2,860,993.11

4,572,433.18

6,162,400.77

1,341,407.58

2,548,128.58

3,857,099.02

5,184,189.95

Time

3.67

7.56

6.73

7.31

0.42

0.95

1.58

2.34

IT

21

19

20

20

18

21

20

21

NSS

1

1

1

1

1

1

1

1

NLW

1

1

2

2

2

3

3

4

#Warehouses (nW) = 10

OF

1,653,895.92

2,860,993.11

4,572,433.18

6,162,400.77

1,334,676.43

2,523,856.89

3,833,785.20

5,173,655.55

Time

154.73

640.28

2,444.92

4,336.02

15.47

364.33

951.34

2,163.33

IT

137

237

438

476

90

194

264

382

NSS

1

1

1

1

1

1

1

1

NLW

2

1

2

2

2

3

3

5

#Suppliers (nS) = 2

#Warehouses (nW) = 5

OF

902,217.35

1,771,093.71

3,003,583.48

3,928,232.99

1,341,407.58

2,519,762.03

3,685,399.79

4,885,462.65

Time

0.09

0.06

1.55

0.59

0.84

1.02

1.54

1.39

IT

9

9

13

13

22

22

22

19

NSS

2

2

2

2

1

2

2

2

NLW

2

2

2

2

2

4

4

5

#Warehouses (nW) = 10

OF

711,512.07

1,331,319.76

2,339,769.96

2,995,884.98

1,264,357.39

2,237,598.46

3,246,475.83

4,154,951.58

Time

2.16

4.38

138.55

323.36

37.36

347.52

837.80

704.27

IT

23

44

119

144

110

160

246

245

NSS

2

2

2

2

2

2

2

2

NLW

2

2

2

2

3

5

4

6

#Suppliers (nS) = 3

#Warehouses (nW) = 5

OF

902,217.35

1,771,093.71

3,003,583.48

3,928,232.99

1,127,656.11

2,116,536.19

3,269,642.37

4,373,393.58

Time

0.02

0.08

0.72

0.34

0.30

0.28

0.61

1.73

IT

9

9

13

13

16

15

18

17

NSS

2

2

2

2

1

3

3

3

NLW

2

2

2

2

2

4

4

4

#Warehouses (nW) = 10

OF

711,512.07

1,331,319.76

2,339,769.96

2,995,884.98

1,064,691.21

1,884,622.88

2,888,657.25

3,687,404.66

Time

1.72

4.61

161.25

312.48

11.09

87.61

598.34

393.86

IT

24

43

116

146

70

122

203

160

NSS

2

2

2

2

2

3

3

3

NLW

2

2

2

2

3

5

4

5

#Suppliers (nS) = 4

#Warehouses (nW) = 5

OF

902,217.35

1,771,093.71

3,003,583.48

3,928,232.99

1,114,799.06

2,080,184.58

3,233,290.76

4,329,027.10

Time

0.05

0.11

0.56

0.55

0.28

0.16

0.89

1.22

IT

9

10

13

13

14

15

18

17

NSS

2

2

2

2

2

4

4

4

NLW

2

2

2

2

2

4

4

5

#Warehouses (nW) = 10

OF

711,512.07

1,331,319.76

2,339,769.96

2,995,884.98

1,051,834.15

1,849,241.84

2,844,935.85

3,635,912.71

Time

2.64

6.67

171.83

394.81

8.92

60.72

467.98

319.95

IT

27

42

130

162

63

110

175

144

NSS

2

2

2

2

3

4

4

4

NLW

2

2

2

2

3

5

4

5

#Suppliers (nS) = 5

#Warehouses (nW) = 5

OF

796,223.69

1,588,750.60

2,715,683.21

3,534,363.11

1,114,799.06

2,064,311.57

3,182,573.28

4,260,916.25

Time

0.16

0.44

0.41

0.89

0.19

0.31

0.64

0.77

IT

8

9

11

13

14

15

16

15

NSS

2

2

2

2

2

4

4

4

NLW

2

2

2

2

2

4

4

5

#Warehouses (nW) = 10

OF

686,625.76

1,295,891.76

2,246,120.38

2,883,470.01

1,051,834.15

1,847,216.18

2,837,172.63

3,635,912.71

Time

2.38

4.25

105.48

239.78

12.80

64.20

498.11

435.16

IT

27

41

108

141

70

113

178

157

NSS

3

3

3

3

3

4

4

4

NLW

3

3

4

4

3

5

5

5

 

Base Instance 3

Base Instance 4

#Customers (nC)

#Customers (nC)

5

10

15

20

5

10

15

20

#Suppliers (nS) = 1

#Warehouses (nW) = 5

OF

995,112.73

2,515,167.29

4,200,934.27

5,258,949.57

1,238,599.35

2,832,975.75

4,325,107.64

5,424,879.07

Time

0.16

0.44

0.75

2.03

0.30

0.14

1.20

1.75

IT

12

16

19

19

13

13

17

17

NSS

1

1

1

1

1

1

1

1

NLW

1

2

2

2

2

2

2

2

#Warehouses (nW) = 10

OF

947,693.14

2,429,290.47

4,037,254.73

5,065,233.24

1,057,054.89

2,046,949.19

3,262,833.68

4,198,252.61

Time

3.92

367.42

1,088.72

1,667.41

10.47

38.22

254.28

352.20

IT

63

208

296

320

73

84

137

160

NSS

1

1

1

1

1

1

1

1

NLW

2

2

2

3

2

2

2

2

#Suppliers (nS) = 2

#Warehouses (nW) = 5

OF

911,715.73

2,339,536.84

3,759,968.83

4,775,702.42

1,104,478.66

2,715,635.28

4,066,848.29

4,977,718.31

Time

0.11

0.95

0.59

1.14

0.72

0.98

3.28

1.34

IT

13

19

17

18

22

22

24

22

NSS

2

2

2

2

2

2

2

2

NLW

2

3

3

3

3

4

4

4

#Warehouses (nW) = 10

OF

872,213.99

2,250,428.74

3,618,518.00

4,606,189.29

966,004.74

1,982,957.17

3,146,368.06

3,959,410.78

Time

4.48

415.91

567.47

782.55

15.53

46.00

305.33

254.34

IT

59

189

184

204

88

97

173

163

NSS

2

2

2

2

2

2

2

2

NLW

3

3

3

3

3

3

3

3

#Suppliers (nS) = 3

#Warehouses (nW) = 5

OF

712,827.76

1,802,342.72

2,797,581.10

3,669,264.02

1,104,478.66

2,715,635.28

4,066,848.29

4,977,718.31

Time

0.09

0.14

0.30

0.44

0.77

1.05

2.41

1.80

IT

9

12

12

12

22

21

22

21

NSS

2

2

2

3

2

2

2

2

NLW

2

2

2

3

3

4

4

4

#Warehouses (nW) = 10

OF

646,117.64

1,611,471.85

2,551,847.98

3,369,148.32

923,324.60

1,939,914.68

2,914,593.43

3,715,791.39

Time

2.97

27.83

43.02

139.97

14.38

36.25

35.56

44.16

IT

32

70

82

103

76

85

77

71

NSS

2

2

2

2

3

2

3

3

NLW

4

4

4

4

3

3

3

3

#Suppliers (nS) = 4

#Warehouses (nW) = 5

OF

712,827.76

1,802,342.72

2,797,581.10

3,669,264.02

1,028,530.32

2,464,786.74

3,704,009.89

4,617,788.52

Time

0.05

0.19

0.33

0.16

0.48

1.31

1.50

2.38

IT

9

12

12

12

19

17

19

20

NSS

2

2

2

3

2

3

3

3

NLW

2

2

2

3

3

4

4

4

#Warehouses (nW) = 10

OF

646,417.64

1,611,471.85

2,551,847.98

3,369,148.32

691,545.15

1,318,757.45

2,018,605.18

2,610,180.59

Time

2.75

26.53

44.48

143.33

2.81

4.58

8.33

15.55

IT

32

70

82

103

40

51

47

44

NSS

2

2

2

2

3

3

3

3

NLW

4

4

4

4

3

3

3

3

#Suppliers (nS) = 5

#Warehouses (nW) = 5

OF

712,827.76

1,802,342.72

2,797,581.10

3,669,264.02

997,555.19

2,359,739.83

3,574,912.38

4,482,864.95

Time

0.05

0.33

0.34

0.27

0.28

0.70

1.91

1.69

IT

9

12

12

12

17

18

20

18

NSS

2

2

2

3

3

3

3

3

NLW

2

2

2

3

3

3

3

4

#Warehouses (nW) = 10

OF

641,973.51

1,610,049.60

2,538,434.70

3,357,335.96

691,545.15

1,318,757.45

2,018,605.18

2,610,180.59

Time

2.91

29.30

52.64

130.23

2.72

5.88

7.22

16.08

IT

32

72

77

104

43

49

40

43

NSS

3

3

3

3

3

3

3

3

NLW

4

5

5

5

3

3

3

3

 

Base Instance 5

#Customers (nC)

5

10

15

20

#Suppliers (nS) = 1

#Warehouses (nW) = 5

OF

2,178,244.28

4,328,128.83

6,116,851.14

8,124,073.96

Time

0.53

1.77

7.83

6.84

IT

15

18

19

19

NSS

1

1

1

1

NLW

1

2

2

2

#Warehouses (nW) = 10

OF

1,663,819.94

3,331,834.97

4,812,300.25

6,098,933.05

Time

29.98

675.22

1,665.34

1,786.20

IT

94

260

348

311

NSS

1

1

1

1

NLW

2

3

4

4

#Suppliers (nS) = 2

#Warehouses (nW) = 5

OF

1,935,285.14

3,788,038.48

5,311,398.25

7,204,257.02

Time

0.66

3.77

7.09

4.11

IT

19

23

23

22

NSS

1

1

1

1

NLW

2

2

2

2

#Warehouses (nW) = 10

OF

1,293,913.57

2,617,644.85

3,838,034.86

4,996,407.90

Time

9.78

159.97

502.75

457.14

IT

52

95

172

146

NSS

2

2

2

2

NLW

2

3

4

4

#Suppliers (nS) = 3

#Warehouses (nW) = 5

OF

1,140,038.99

2,283,616.46

3,058,355.76

4,034,370.07

Time

0.20

0.50

0.48

0.58

IT

10

15

15

15

NSS

1

1

1

1

NLW

1

2

2

2

#Warehouses (nW) = 10

OF

1,076,136.77

2,137,499.81

2,860,586.43

3,696,629.23

Time

3.38

65.48

194.55

60.67

IT

32

98

115

102

NSS

2

3

3

3

NLW

2

3

3

3

#Suppliers (nS) = 4

#Warehouses (nW) = 5

OF

1,140,038.99

2,255,318.32

3,015,394.18

3,991,818.42

Time

0.08

0.41

0.92

1.66

IT

10

15

17

17

NSS

1

2

2

2

NLW

1

2

2

2

#Warehouses (nW) = 10

OF

964,500.00

1,925,217.76

2,593,771.64

3,430,290.72

Time

1.41

9.94

24.81

9.19

IT

20

53

57

51

NSS

2

3

3

3

NLW

2

3

3

3

#Suppliers (nS) = 5

#Warehouses (nW) = 5

OF

1,140,038.99

2,255,318.32

3,015,394.18

3,991,818.42

Time

0.06

0.56

1.69

1.41

IT

10

15

17

17

NSS

1

2

2

2

NLW

1

2

2

2

#Warehouses (nW) = 10

OF

964,500.00

1,925,217.76

2,593,771.64

3,430,290.72

Time

1.78

10.31

26.23

9.16

IT

20

53

57

51

NSS

2

3

3

3

NLW

2

3

3

3

Appendix D

 

AVERAGE

#Customers (nC)

5

10

15

20

#Suppliers (nS) = 1

#Warehouses (nW) = 5

OF

1.481.451,97

3.017.078,71

4.614.485,05

6.030.898,67

Time

1,02

2,17

3,62

4,06

IT

15,80

17,40

19,00

19,20

NSS

1,00

1,00

1,00

1,00

NLW

1,40

2,00

2,20

2,40

#Warehouses (nW) = 10

OF

1.331.428,07

2.638.584,93

4.103.721,41

5.339.695,04

Time

42,92

417,09

1.280,92

2.061,03

IT

91,40

196,60

296,60

329,80

NSS

1,00

1,00

1,00

1,00

NLW

2,00

2,20

2,60

3,20

#Suppliers (nS) = 2

#Warehouses (nW) = 5

OF

1.239.020,89

2.626.813,27

3.965.439,73

5.154.274,68

Time

0,48

1,36

2,81

1,72

IT

17,00

19,00

19,80

18,80

NSS

1,60

1,80

1,80

1,80

NLW

2,20

3,00

3,00

3,20

#Warehouses (nW) = 10

OF

1.021.600,35

2.083.989,80

3.237.833,34

4.142.568,91

Time

13,86

194,75

470,38

504,33

IT

66,40

117,00

178,80

180,40

NSS

2,00

2,00

2,00

2,00

NLW

2,60

3,20

3,20

3,60

#Suppliers (nS) = 3

#Warehouses (nW) = 5

OF

997.443,77

2.137.844,87

3.239.202,20

4.196.595,79

Time

0,28

0,41

0,90

0,98

IT

13,20

14,40

13,80

15,60

NSS

1,60

2,00

2,00

2,20

NLW

2,00

2,80

3,00

3,00

#Warehouses (nW) = 10

OF

884.356,46

1.780.965,80

2.711.091,01

3.492.971,72

Time

6,71

44,36

206,54

190,23

IT

46,80

83,60

118,60

116,40

NSS

2,20

2,40

2,60

2,60

NLW

2,80

3,40

3,20

3,40

#Suppliers (nS) = 4

#Warehouses (nW) = 5

OF

979.682,70

2.074.745,21

3.150.771,88

4.107.226,21

Time

0,19

0,43

0,84

1,19

IT

12,20

13,80

15,80

15,80

NSS

1,80

2,60

2,60

2,80

NLW

2,00

2,80

2,80

3,20

#Warehouses (nW) = 10

OF

813.161,80

1.607.201,73

2.469.786,12

3.208.283,47

Time

3,71

21,69

143,49

176,57

IT

36,40

65,20

98,20

100,80

NSS

2,40

2,80

2,80

2,80

NLW

2,80

3,40

3,20

3,40

#Suppliers (nS) = 5

#Warehouses (nW) = 5

OF

952.288,94

2.014.092,61

3.057.228,83

3.987.845,35

Time

0,15

0,47

1,00

1,00

IT

11,60

13,80

15,20

15,00

NSS

2,00

2,60

2,60

2,80

NLW

2,00

2,60

2,60

3,20

#Warehouses (nW) = 10

OF

807.295,71

1.599.426,55

2.446.820,91

3.183.438,00

Time

4,52

22,79

137,94

166,08

IT

38,40

65,60

92,00

99,20

NSS

2,80

3,20

3,20

3,20

NLW

3,00

3,80

4,00

4,00

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Tapia-Ubeda, F.J., Miranda-Gonzalez, P.A. & Gutiérrez-Jarpa, G. Integrating supplier selection decisions into an inventory location problem for designing the supply chain network. J Comb Optim 47, 2 (2024). https://doi.org/10.1007/s10878-023-01100-y

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